Unit 07

Department of Communication Engineering, NCTU 1
Unit 7 Multi-Level Gate Circuits/
NAND and NOR Gates

7.1 Multi-Level Gate Circuits

Logic Design Unit 7 Multi-Level Gate Circuits Sau-Hsuan Wu
 The maximum number of gates cascaded in series
between a circuit input and the output is referred to as the
number of levels of gates
 ANR-OR circuit  A level of AND gates followed by a
OR at the output
 OR-AND circuit  A level of OR gates followed by a
AND at the output
 OR-AND-OR circuit  A level of OR gates followed by a
level of AND gates followed by
OR gate at the output
 A function written in SOP form or in POS form
corresponds to a two-level gate circuit
 Inverters which are connected directly to input variables
will not be counted when determining the # of levels

 Example: A four-level realization with 6 gates and 13
gate inputs

 Another realization of 3 levels of gates. There are six
gates and 19 gate inputs in total

 Example: Find a circuit of AND and OR gates to realize
 By Karnaugh map, f = a’c’d + bc’d + bcd’+ acd’(7-1)
( , , , ) (1,5,6,10,13,14)f a b c d m

 Factoring (7-1) yields f=c'd(a'+b)+cd'(a+b) (7-2)
 Both realizations use 5 gates, but the later one has fewer
inputs with on more level of gate delays

 An alternative realization in POS form: obtained from the
0’s on the Karnaugh map
f'=c'd'+ab'c'+cd+a'b'c (7-3)
f=(c+d)(a'+b+c)(c'+d')(a+b+c') (7-4)

 Partially multiplying out (7-4) using (X+Y)(X+Z)=X+YZ :
f =[c+d(a'+b)][c'+d'(a+b)] (7-5)
=(c+a'd+bd)(c'+ad'+bd') (7-6)
 Eq. (7-6) leads to a 3-level AND-OR-AND circuit

 Summaries:
 If an expression for f’has n-levels, the complement of that
expression is an n-level expression of f
 To realize f as an n-level circuit with an AND-gate output,
one procedure is to find an n-level expression for f’with an
OR operation at the output and then complement the
expression for f’

7.2 NAND and NOR Gates

 NAND and NOR gates are frequently used because they
are generally faster and use fewer components than AND
or OR gates
 Any logic function can be implemented using only
NAND or only NOR gates
 An n-input NAND gate is
1 2 1 2( )n nF X X X X X X      

 Similarly, an n-input NOR gate is
 A set of function is said to be functionally complete if any
Boolean function can be expressed in terms of this set of
operations, e.g. AND, OR and NOT
 Any set of logic gates which can realize AND, OR, and
NOT is also functionally complete
1 2 1 2( )n nF X X X X X X        

 E.g. AND and NOT form a functionally complete set of
gates, since
 NAND is also functionally complete

7.3 Design of Two-Level Circuits
Using NAND and NOR Gates

 A two-level circuit composed of AND and OR gates is
easily converted to a circuit composed of NAND gates or
NOR gates. E.g. converting from a minimum SOP
 (7-13)：AND-OR
(7-14)：NAND-NAND
(7-15)：OR-NAND
(7-16)：NOR-OR

 Obtaining a two-level circuit containing only NOR gates
should start with the minimum POS for F, instead of SOP
 E.g.
 (7-18)：OR-AND
(7-19)：NOR-NOR
(7-20)：AND-NOR
(7-21)：NAND-AND

 Two of the most commonly used circuits are the NAND-
NAND and the NOR-NOR
 Procedure for designing a min 2-level NAND-NAND
circuit
 Find a minimum SOP for F
 Draw the corresponding two-level AND-OR circuit
 Replacing all gates with NAND gates

 Procedure for designing a min 2-level NOR-NOR circuit
 Find a minimum POS for F
 Draw the corresponding two-level OR-AND circuit
 Replace all gates with NOR gates

7.4 Design of Multi-Level NAND-
and NOR-Gate Circuits

 The following procedure may be used to design multi-
level NAND-gate circuits
 Simplify the switching function to be realized
 Design a multi-level circuit of AND and OR gates.
 The output gate must be a OR gate
 AND-gate outputs cannot be used as AND-gate inputs; OR-
gate outputs cannot be used as OR-gates inputs
 Replace all gates with NAND gates
 The procedure for the design of multi-level NOR-gate
circuits is exactly the same as for NAND-gate circuits
except that the output gate of the circuit must be an AND
gate, and all gates are replaced with NOR gates

7.5 Circuit Conversion Using
Alternative Gate Symbols

 Alternative representations for an inverter
 Alternative representations for AND, OR, NAND and
NOR gates

 The procedure for converting AND-OR circuit to a
NAND or NOR circuit
 Convert all AND gates to NAND gates by adding an
inversion bubble at the output
 Convert all OR gates to NAND gates by adding inversion
bubbles at the inputs
 Whenever an inverted output drives an inverted input, these
two inversions cancel
 Whenever a noninverted gate output drives an inverted gate
input or vice versa, insert an inverter so that the bubble will
cancel

 Example

7.6 Design of Two-Level, Multiple-
Output Circuits

 Solution of digital design problems often requires the
realization of several functions of the same variables. The
use of some gates in common between two or more
functions sometimes leads to a more economical circuit
 E.g. we have

 The individual realizations to them are

 Observe that the term ACD is necessary for the
realization of F1 and that A’CD is necessary for F3. If
replacing CD in F2 by A’CD + ACD, the realization of
CD is unnecessary

 In realizing multiple-output circuits, the use of a
minimum sum of prime implicants for each function does
not necessarily lead to a minimum cost solution
 When designing multiple-output circuits, try to minimum
the total number of gates required
 E.g.
1
2
3
(2,3,5,7,8,9,10,11,13,15)
(2,3,5,6,7,10,11,14,15)
(6,7,8,9,13,14,15)
f m
f m
f m







 The corresponding Karnaugh maps are

 By inspection, we can see that
 a’bd from f2, abd from f3 and ab’c’from f3 can be used in f1.
Replacing bd with a’bd + abd, the gate needed to realize bd
can be eliminated.
 m10 and m11 in f1 are already covered by b’c, and ab’c’from
f3 can be used to cover m8 and m9, thus ab’being eliminated
 The minimum solution is therefore

 Determination of essential prime implicants for multiple-
output realization
 The prime implicants essential to an individual function
may not be essential to the multiple-output realization
 bd is an essential prime implicant of f1 but not of all f’s

 When searching for a prime implicant to an multiple-output
realization,
 Check each 1 which do not appear on the other function maps
 Example 1
 c’d is essential to f1, bd’is essential to f2
 abd is not essential since it appears on both maps

 Example 2
 a’d’and a’bc’are essential to f1
 bd’and a’b’c are essential to f2

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